Structures, Failure, Inference and Prediction

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STRUCTURES, FAILURE, INFERENCE AND PREDICTION M. B. Beck Departll/fllt of Cil'il Engineering, llllpmal Co/lege, LOIII/on .'1\\'/ 2BL', ['K

Abstract. The immediate and most difficult problems in analysing and controlling the behaviour of environmental systems are not those concerned with either the estimation of model (polynomial) orders or the convergence properties o f parameter estimation algorithms. This paper examines therefore the problem of model structure ~entification, formulating it first as a problem of exposing the failure of inadequate ,~~ypotheses, and then as a problem of drawing inferences in order to generate ' alternative, improved hypotheses. Two possible modifications of existing identification ,~ethods are outlined, one based on the graph representations of engineering structural analysis, the other being akin to the heuristics of expert systems. Lastly, since the purpose of a model of an environmental system is rarely that of closed-loop control, the paper discusses the problems of ambiguity and contradiction in making predictions ' of future behaviour. Keywords. Filtering; graph theory; heuristics; identification; prediction; water pollution.

INTRODUCTION

environmental systems is inherently and significantly imprecise, then to seek algebraic 'or differential-equation models of such systems may be quite misguided (Jowitt, 1984).

Conventional wisdom on identifying the behaviour of environmental systems is under attack. For example, consider a typical environmental problem such as that of surface water acidification, i.e.

Set against the richness of these pragmatic observations the directions of theoretical progress can seem sadly out of alignment. Model structures, their failure, the drawing of inferences about anomalous behaviour, and making predictions without a view to closed-loop control, are not subjects that would usually be considered the essence of system identification. Yet they are crucially important to the analysis and management of environmental systems, and in that context largely ignored by control theory and cybernetics.

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little, incoherent prior theory, together with intensive time-series observations of behaviour in at most one or two specific hydrological catchments over one or two years, Determine the causal mechanisms of surface water acidification and make predictions of the likely responses of many catchments over many years. And now consider whether customary interpretations of system identification have any role to play in solving this and similar problems. Unequivocally they should; but can they, for certain observations suggest they cannot: (i) Conventional methods of system identification, such as those o~ recursive estimation, perform well on well-posed problems of single input/single output systems where int"itively obvious input/ output cause-effect perturbations are dominant, though not exclusive (Beer and Young, 1983); but they are not especially effective for multivariable problems with obscure and nonlinear causal mechanisms (Beck, 1983). (ii) When the available observations are sparse, as so often is the case, conventional methods are a priori inapplicable. Under these circumstances a form of "speculative simulation modelling" proposed by Hornberger and Spear (1981) is highly appropriate; indeed, the argument that it is applicable more generally is persuasive and undoubtedly a challenge to conventional approaches (Fedra, 1983; Hornberger and Cosby, 1985). (iii) If prior theoretical knowledge and empirical observation of the behaviour of

STRUCTURES The identification of environmental systems rarely begins with a given model structure, neither is model structure identification simply a matter . of estimating the orders of polynomials in the backward-shift operator, nor does interest end with the asymptotic convergence of the parameter estimates. It has hitherto been the paradigm to develop as "comprehensive" a "physics-based", "mechanistic" representation of the system as possible (Park and co-workers, 1974; Ch en and Smith, 1979 ) . This view still prevails (Goldstein and co-workers, 1984; Carmichael, 1985), although not in so dogmatic a form as previously.

In this paper models will be viewed rather as formalised archives of hypotheses or as vehicles for the analysis of data, a view that emphasises models as means to an end, and no t as ends in themselves. It is convenient then to d~stinguish between three classes of models:

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1-141

M. B. Beck

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ConceFtual analog of model structure: the states (x) are represented as the (fixed) node; and the parameters (~) as the flexible connections between these nodes.

(3)

Here x -is the state vector, ~ a vector of observed input disturbances, y a vector of output response observations (which we assume applies to all classes of model), and e, ~, and B appropriate vectors of model parameter;. The dot notation in (2a) denotes differentiation with respect to time t, the prime notation in (2b) differentiation with respect to a single spatial direction z, ~ is a vector representing the three spatial directions,and tk is the kth sampling instant.

In general class I models, while they may be thought desirable as formal, comprehensive syntheses of prior hypotheses, cannot.be engaged as practical vehicles for the interpretation of field data. Their use implies a principle of a priori inclusion in the model structure of all the hypotheses of conceivable relevance, with subsequent rejection of only those hypotheses that are demonstrably inadequate. The essential difficulty is that it is likely to be virtually impossible to resolve which, among the many hypotheses are inadequate. On the other hand, the class III models, which can so easily be mobilised for the analysis of time-series observations, are not a satisfactory end-point in understanding a system's behaviour. It is instinctive for the scientist to ask how and ~ certain types of behaviour or anomalies are observed, not merely to accept that they are. For instance, a linear (regression) relationship identified between the concentrations of aluminium and hydrogen ions in an upland stream will prompt marty questions, since it runs counter to the expected deductions from chemical equilibrium theory (tfuitehead and co-workers, 1984). The interpretation of anomalies, and the revision of inadequate hypotheses, are not matters that can be resolved without some form of recourse to the archive of hypotheses associated with a class I model. The primary questions of model structure identification are therefore: (i) how to expose the failure (inadequacy) of the component hypotheses of a model structure, for which purposes the conceptual analog of Figure 1 for a class 11 model will be introduced; (ii) how to infer the form of an improved model structure from diagnosis of the failure of an inadequate structure and from the prior knowledge associated largely with the class I model. In answering these questions the class 11 models, especially of the form of (2a), play a central r o le in the rigorous and exhaustive interpretation of field data. Model structure identification is here understood as determining a choice of suitable state vector, an appropriate parameterisation ('::.), and identification of the function f.

FAILURE A simple illustration of the exposure of an inadequate model structure is given in Figure 2 for recursive estimates generated from an extended Kalman filter (EKF); this particular application refers to a two-state (class 11) model of organiC waste degradation in a stretch of river (Beck and Young, 1976). The "deflection" in the recursive estimates &1 (tkitk) over the central period of the record is indicative of an inadequate structure that does not match observed perturbations in the field data, and these perturbations are due -- so it is hypothesised -- to the activities of an algal population (Beck, 1983). Such an analysis is not aimed at identifying the order of a dynamic input/output transfer-function model. Rather, its purpose is to establish the "success" or failure of hypotheses about the description of internal mechanisms of behaviour. There will be many constituent hypotheses, even in a modestly complex case, and these will be bound inseparably to form the model structure as a whole. The questions of interest are: has a failure occurred, what is the connection between the failed hypothesis and the mismatch between the model and the observations, and what are the relative weaknesses/strengths of the individual hypotheses? The following is a conjecture on how to generate responses to these questions.

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20

40

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(days)

Failure of an inadequate model structure: (a) state estimates XI (tkitk) and observations (dots) (b) parameter estimates UI(tkitk)'

Structures , Fail ure, I nference and Prediction An Analogy with

Physic~l

Engineering Structures

The key to th~ analogy is Figure 1. The nodes of the graph (or ', model structure) repr esent the model's stat, variables and the ~r~nches (structural members) the model parameters. The external loads placed on the structure are assumed to be equivalent to the innovations process error sequences o f a filter - like algorithm. The "distortion" necessary for the model structure to be matched with the structure of the dynamics underlying the observations is reflected in the deflections of the recursive parameter estimation trajectories (as in Figure 2). The capacity of a structural member to resist deformation, i.e. its mechanical propert i es, corresponds in some way to the confidence attached to a component hypothesis (as pa rameter ised through ~) . The original motive for this conceptual analogy was the need simply to define the problem of model structure identification. It then appeared to correlate w.ell with the notions of structural failure~d col lapse , and thus far i t has dictated the log~cal distinction between the falsification of bold, ,O nfident hypotheses and speculation about relatively uncertain hypotheses (Beck, 1983; 1985a). Now we may note further that, first, graph s , networks and connectivity are intimately related to the analysis of structural identifiability, especially in th e adjacent subjects of pharmacokinetic s and ecological systems (Cobelli and co-workers, 1979) . Second, much of the ba s is of structural mechanics is founded upon graph theory and network representations (Spillers, 1972). Third, within structural mechanics plastic limit ana l ysis facilitates (for the specification of our present problem) access to the concepts and results of (see , for example, f1unro and Smith, 1972): (i) constraints that crisply distinguish failure fr om non- failure; (ii). constrained optimisation, with an accompanying scalar measure of performance that implies distance from an "optimum" (structure?) ; (iii) dual linear programs, from which can be obtained a ranking of constraint (failure?) sensitivities. Fourth, and possibly of most immediate significance, there is the duality of nodal and mesh descriptions of a network . Thus, for the network of Figure 1 a nodal description ought to correspond with a description of the system's behaviour in terms of the state dynamics, and a mesh description should correspond I
..

This duality, which can be interpreted as the duality between state estimation given "perfectly known" parameters and parameter estimation given "perfectly known" states, could be crucial to the development of novel algorithms of model structure identification (Beck, 1985b). In particular, an innovations process representation has been shown to be important for both problems (Ljung, 1979; Young, 1979), which would suggest dual descriptions o f the forms (state descrip ti on)

R(tk+lltk)=~x(tkltk_l)+r~(tk)+K~(tkltk_l) (4a) (parameter description)

~(tk+lltk)= ~(tkltk_l) + L~* (tklt k _ l )

problems are then: (i ) In (4a) assume x to be "known perfectly", then estimate the elements of ~ , r, and the gain K; (ii) In (4b) assume ~ to be "known perfectly", then estimate x and the elements of the gain L . It is moreover possible to visualise a practical scheme for alternate solution of these dual problems along the lines of the iterat i verecursive algorithms discussed in Yo ung ( 1984 ) '. For model structure identification problem (ii) has the greater significance. Since ~ is assumed to be known perfectly this satisfies the requirement of seeking to falsify bold , confidently-stated hypotheses (i.e. the collapse o f a structure whose members have little tensile strength). Furthermore, the gain L can be seen in Figure 3 to possess important properties as a mechanism for "distributing" the effects of mismatches among the component members of the model structure. In other words, knowledge of L may assist in answering the question: to the failure of which hypotheses is a specific mismatch between the model and the observations due? The estimation of L is of interest in its own right, and not merely as a means to the improved convergence of the EKF, for example , as a parameter estimator (Ljung , 1979). Hi smatch (load) distribution Innovations (misma tches) v* ~------~

GAIN L

Fig.3.

Extension of the conceptual ana l og of model structure: importance of the gain L in relating mismatches to the failure of constituent model hypotheses .

I NFERENCE The logical connection between an identified anomaly and a failed hypothesis has a special significance in drawing inference about the possible form of an improved model structure from the failure of an inadequate structure. Given our interpretation of the problem of model structure identification there is no systematic "algorithm" for changing an inadequate structure that is equivalent to increasing a polynomial order from n, say, to (n+l) , as would be possible for a class III model structure. And i f no such algorithm obtains , are there (indeed, should there be) heuristics that will suffice? The following is a speculative response, its purpose being to suggest generalisati o n from particular examples o f making the required form of inference. Consider the problem of wind-induced resuspension of sediment material in a shallow lake for which an appropriate (c las s 11) model structure is: i(t) = - fx(t) + gu(t)

(Sa)

(4b)

where V and V* are the innovations sequences for

the two representations and K and L a re gain matrices. The corresponding dual estima tion

1445

(Sb) where u represents the (input) wind veloc ity, x a depth-averaged concentration o f suspended s ol ids

:--1. B, Bel \.;.

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(SS) I f is a settling-rate constant, 9 a parameter associated \v'ith particle resuspension, and h a "background " SS concentration which would be observed in the a bsence o f any wind disturbance. The model in fact refers t o a case study of Lake Balaton, Hungary (Somlyody and co -workers , 1983), and its identificati o n is discussed elsewhere (Beck , 1985a, b). The lake is of an elongated shape, and the experi~ental work was carried out at a pOint locati on r ough ly midway along the lake. Note that the model essent i ally accounts only for two hypotheses (on particle settling and resuspension), although nany other phenomena will in practice influence the observed SS concentration. The model can be derived straightforwardly from a class I representation.

from the possible usefulness of the conjecture. Temperature and background SS concentration could be related either as cause and effect, respectively , or as correlated effec ts o f seasonal

changes in day - length . Average diurnol wind patterns are indeed different f or the first 100 and last 40 days of the record. Yet this could all be spurious reasoning, because h is precisely

that part of the model assumed a priori to account for factors other than particle settling and wind- induced particle resuspension. Temperature

(oC) 24

On identification it is found that the structure of equation (S) fails to characterise observed behaviour over the last 40 days or so of the experimental record in that the model persistently over-estimates the observed SS concentration.

16

It

is perhaps obvious to speculat,e that this anomaly may be due to variations of wind direction (not

8

accounted for in u), and that a form of "forward"

reasoning from the body of a priori knowl~ would run as follows : "IF " wind fetch-length increases (decreases) "THEN " water surface shear stresses increase (decrease) "AND" sediment shear stresses increase (decrease) "AND" more (fewer) sediment particles are (A) resuspended.

Further identificati on shows that wind direction is in fact important, but not in the expected sense of this reasoning;

i.e . not in terms of the

longitudinal component o f the wind (acti ng along the length of the lake) . Prompted by the original anomaly , re-appraisal of the wind - direction data reveals that the dominant wi nds over the last 40 days of the record are from a longitud ina l directi on (in contradistinction to the wi nd pattern for the remainder of the record) and that it is therefore the relatively low transversal wind component that replicates the required decreased rate of sediment resuspension (or increased rate of sediment settling). In short a more complex anomaly has been identified. Equation (Sa) can be integrated over the sampling interval to give (6)

where now u * represents the absolute value of the transversal wind velocity and y* is to be interpreted as the observed ss concentration at anyone of five depths in the water column . Figure 4 shows a comparison between o0ser ved temperature variations in the lake and a recursive estimate of

the parameter h in equation (6) , which represents the background SS concentration at a depth of some 0.3m above the bed of the lake (Chan , 1984). From this suggested corre l a tion between data and a parameter estimate a form of "backward"

reasoning might take place: "IF" h(tk ! t 'ITHEN"

k

) is correlated with temperature temperature influences viscosity which influences particle movement through wateri

" OR" temperature is in fluence d by solar radiation, "hich is influenced by daylength; and day-length influences diu r nal \vind patterns.

(B)

In fact there is ambiguity and even contradiction in this reasoning, although that does not detract

Fig.4.

o 40 80 120 (days) Compa ri son of recursive estimates h (eqn. 6) and observed temperature (dashed line).

In spite of the imperfections in the logic of the above examp l es , certain more general pr i nciples in the heuristics are apparent . First, the reason i ng is drive n by the identification of an anomaly; depends c r ucia l ly o n the corre l ation of several different types of information; a nd util i ses a r api d qua litati ve p r esentation and evaluati on of c h a ins of cause-e f fect hypothese s (without recourse to the ma the mat i cal fo r m of th e p ri o r h ypotheses i n a class I mode l structure ; s ee also related wo rk by Kuipe r s and Kassirer, 1983) . Second , the gen e ratio n of possible exp l ana tions seems to depend upon diversification and disaggregat i on (as in statement B) ; generating plausible explanati ons might conversely depend upon focussing , selection, and aggregation (as in statement A); and the identification of anomalies can focus on aggregate trends (as in Figure 4) or on specific events. Lastly , aggegration and disaggregat i o n suggest a hierarchical format for the reason i ng process, in which respect there are strong simi larities between the present work and that on signal - to - symbol transformation reported by Nii and co - workers (1982). Their objective was to generate a " current best hypothesis " about the state of a complex shipping situation; ours here is to generate a "good hunch" on how to go about improving an inadequate model structure . PREDICTION Because of the dominant paradigm in the development of models of environmental systems it is not uncommon to encounter serious difficulties in a lack of identifiability or an over parameterisation of the more "comprehens ive" class I model structures. It is difficult to resolve which of the constituent model hypotheses have failed -- all the recursive parameter estimates may tend to vary "ith time -- and there is ambiguity in interpreting the nature of the system's past behaviour . The model may contain surplus content (Young , 1978), i . e. descriptions either of a type of behaviour not actually observed in the particular sample of data, or of multiple types of behavi our , the individual components of which cannot be disentangled from

Structures, Failure, Inference and Prediction observations of their collective effect. It is well known that the a posteriori variancecovariance matrix'of parameter estimation errors, P, is indicative:of a lack of identifiability and that the properties of this matrix can be used in solVing problems of model order estimation (Young and co-w~rkers, 1980). In terms of the conceptual analog of Figure 1, P can be thought of as a synoptic measure of the distortion of the model structure brought about in fitting the model to the data. Undesirable though a lack of identifiability may be, in a control -situation it may not be of primary concern. The consequences of the ambiguity do not need to be propagated across a long sampling interval; providing the output stays within the bounds of previously identified modes of behaviour the ambiguity may be neither apparent nor detrimental; and should other modes of previously unidentified behaviour be excited, the adaptive controller ought quickly to be able to resolve the ambiguity. This is not the case for the problem of prediction. While it ~~ always be the goal to seek to eliminate ambtguity (and contradiction) during identificatft:in, it may well be that i t is intrinsic to prediction, and moreover that it has a useful role to play. The interesting problems are usuallY , those of predicting different forms of behaviour under different circumstances and at different points in space and time, And if such behaviour is truly different from that observed in the past, then it will either not be included in the model or be included as surplus content. This being the case, the errors associated with a state variable prediction that are propagated from the posterior covariance matrix P (among other sources) assume considerable significance (Beck, 1983). Suppose, for example, that a model has been fitted toa set of data, that the model structure suffers from problems of identifiability, and that many combinations of parameter estimates give "acceptably good" fits to the data. Is that model capable of generating ambiguous or contradictory predictions of the future, and how might these terms be defined? There is therefore an important logical connection between identification and prediction; a form of logic that not only works both ways, but may also make a virtue of demonstrating the lack of identifiability of a constituent model hypothesis. Figure 5 shows two sets of predictions and prediction error variances corresponding to two equally possible combinations of model parameter values. It is assumed that the errors are normally-distributed and that each prediction is based on the same P matrix. The results in fact refer to the behaviour of a phytoplankton population in a model of Lake Ontario (Beck and Halfon, 1985). An intuitive definition of ambiguity might be that the mean of one prediction does not lie "'ithin one standard deviation (0) of the other prediction; contradiction would be the case where (X\+OI) «x 2 -a 2 ) ; and the predictions would otherwise be termed indistinct. On these bases Figure 5 illustrates contradictory rather than ambiguous predictions. Clearly this should provoke some concern at the often asserted claim that a class I model representation is inherently superior in its "predictive power u • It also raises interesting questions about the problem of prediction itself. For instance, instead of asking whether a model can predict radically different behaViour, the question is, if radically different behaviour may occur in the future, can this be shown not to have been identifiable from past observations (otherwise such behaviour cannot be called radically different)?

1447

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Predictions derived from two equally possible combinations of parameter values: A = ambiguous; C = contradictory; I indistinct. The dashed lines denote ± a bounds for each prediction X.

CONCLUSIONS If a technique continues to perform well on an easily definable and consistent problem, there is little incentive to search for substantially different techniques. Quite the opposite is presently the case in the identification of environmental systems. Here convention is subject to challenge: the field observations are sparse; and the systems, it may be argued, are inherently imprecise. These challenges, rather than being rejected (for instance, on the grounds that sparseness of data and imprecision are transient phenomena of new fields of research), are positive stimuli and a potential source of methodological enrichment. In this paper we have examined possible avenues for the development of novel algorithms for model structure identification that would combine the conventions of recursive estimation with certain elements of graph theory and structural mechanics. From this, innovations process representations, and the duality of state estimation given perfect knowledge of the parameters and parameter estimation given perfect knowledge of the states emerge as important guiding principles. A second duality -- not discussed here -- is that between lumped~parameter dynamiC models (equation 2(a» and quasi-steady-state approximations (equation2(b». Access to information about a system via the latter is usually either ignored by or irrelevant to control theory. An understanding of spatial variability can be as important to the analysiS of environmental systems as the quantification of temporal variability (see, for example, Beck, 1985a) . The paper has also examined the possibilities of applying heuristics to the problem of inferring an improved model structure from the failure of an inadequate structure. Doubtless there are several areas that might be amenable to such an approach, an example being the correlation and collation of the information contained in the residual error

~1.

1448 statistics, parameter estimation error variances,

and the parameter and state trajectories generated during identification of the prior model. They would be necessary there not least to avoid being overwhelmed by a surfeit of diagnostics (Beck, 1983). However, the discussion has been focussed instead on the forms of reasoning required to generate and evaluate "good hunches" as to how the model structure might be improved. Such reasoning draws upon: qualitatively stated candidate hypotheses from prior knowledge: reason ing by parallel argument from adjacent subject fields: a unique, possibly idiosyncratic, and certainly not entirely logical association of conceptsi and last,

but not least, serendipity . It seems a contradiction in terms to seek a logic for this last. REFERENCES Beck, 11.B. (1983). Uncertainty, system identification, and the prediction of water quality. In 11.B. Beck and G.van Straten (Eds.), Uncertainty and Forecasting of Water Quality. Springer, Berlin. pp. 3-68 . Beck, M. B. (1985a). Lake eutrophication: identification of tributary nutrient loading and sediment resuspension dynamics . Applied Mathematics and Computation (in press) . Beck, M.B. (1985b). The analysis of environmental t~me-ser~es.

In Proc. Convegno Societa Italiana

di Statistica. Giardini Naxos, Sicily. (in press) . Beck, M. B., and P.C. Young (1976). Systematic identification of DO-BOO Model structure. Proc. Am. Soc. Civil Engrs . , J. Env. Eng. Div . , 102, 909-927. Beck, M.B., and E. Halfon (1985). Prediction and prediction error propagation: a case study of Lake Ontario. (in preparation) . Beer, T . , and P.C. Young (1983) . Longitudinal dispersion in natural streams. J. Env. Engineering, 109 , 1049-1067. ------Carmichael G.R. (1985) . Sources of error and uncertainty in Eulerian long range transport models. In Proc . American Meteorological Society lIeeting on Sources and Evaluation of Uncertainty in Long Range Transport Models. Woods Hole, Massachusetts. (to appear). Chan, S.K . C. (1984) . I~ind-induced resuspension of sediment in a shallow l a ke . M.Sc. Thesis. Department of Civil Engineering, Imperial College, London. Chen, C.I'., and D.J. Smith . (1979) . Preliminary insights into a three-dimensional ecologicalhydrodynamical model. In D. SCdvia and A. Robertson (Eds.) Perspectives in Lake Ecosystem Modelling . Ann Arbor Science, Ann Arbor, Michigan. pp . 249- 279 . Cobelli, C . , A . Lepschy, and G. Romanin - Jacur . (1979). Structural identifiability of linear compartmental models . In E. Halfon (Ed.) Theoretical Systems Ecology. Academic Press, New York. pp. 237-258. Fedra,K. (1983) . Environmental modeling under uncertainty: Monte Carlo Simulation. Research Report RR- 83 - 28 , International Institute for Applied Systems Analysis, Laxenburg, Austria . Goldstein, R. A . , S.A. Gherini, C.W. Chen, L. Mok, and R.J.II. Hudson. (1984). Integrated acidification study (ILWAS): a mechanistic ecosystem analysis. Phil. Trans. R . Soc. London, B 305, 409 - 425. Hornberger, G.M., and B.J. Cosby . (1985). Selection of parameter values in environmental models using sparse data. Applied Mathematics and Computation. (in press) . Hornberger , G. M., and R.C. Spear. (1981). An approach to the preliminary analysis of environmental systems. J . Env. Hgtnt. , 12 , 7-18.

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24, 36 - 50. Munro. J •• and D.L. Smith. (1972). Linear programming duality in plastic analysis and synthesis. In Proc. Symposium on Computeraided Structural Design. Peter Peregrinus, Stevenage. Vol. 1, pp. Al, 22 -A l .54 . Nii, II.P., E.A. Feigenbaum , J.J. Anton. , and A.J. Rockmore. (198 2 ) . Signal-to-symbol transformation: HASP/SlAP case study. The AI Magazine, 3 (Spring), 23-35. PClrk , R.A . , and co-workers . (1974). A generalised model for simulating lake ecosystems. Simulation , 23, 33 -50. Somlyody, L., S. Herodek, and J. Fischer (Eds .) . (1983). Eutr ophication of shallow lakes: modeling and management. Collaborative Proceedings CP - 83-S3, International Institute for Applied Systems Analysis, Laxenburg, Austria .

Spillers, W. P . (1972). Automated Structural Analysis: An Introduction. Pergamon, New York . Whitehead , P.G. , C. Neal, S. Seden-Perriton, N. Christophersen, and S. Langan. (1984). A time series approach to modelling stream acidity. In Proc .. Uppsala Conf. on 110delling Stream Acidification Processes (in press) . Young, P.C . (1978). General theory of modeling for badly defined systems. In G.C.Vansteenkiste (Ed.) Modeling, Identification and Control of Environmental Systems. North-Holland, Amsterdam. pp. 103-135. Young, P . C . (1979)" Self-adaptive Kalman filter. Electronics Letters, 15(2), 358-360. Young, P . C . (1984). Recursive Estimation . Berlin , Springer . Young, P.c., A.J. Jakeman, and R. McMurtrie. (1980). . An instrumental variable method for model order identification. Automatica, 16, 281-294.